Question: A new shopping mall is gaining in popularity. Every day since it opened, the number of shoppers is $5\%$ more than the number of shoppers the day before. The total number of shoppers over the first $10$ days is $1258$. How many shoppers were at the mall on the first day? Round your final answer to the nearest integer.
Notice that the daily counts of shoppers form a geometric sequence. The total number of shoppers after $ n$ days is the ${\text{sum}}$ of the first $n$ terms in the sequence. This is called a geometric series. This is the formula for that sum: $ S={a}\left(\dfrac{1-{r}^{ n}}{1-{r}}\right)$ where ${a}$ is the first term and ${r}$ is the common ratio. We can use this formula, along with the given information, to find how many shoppers were at the mall on the first day, $ a$. Using the given information We are given that the number of shoppers each day is ${5\% \text{ more}}$ than the number of shoppers the day before. So we'll use a common ratio of ${1.05}$ for $ r$. We are given that the total number of shoppers over the first ${10}$ days is ${1258}$. So the sum of series $ S$ is ${1258}$ shoppers, and the number of terms $ n$ is $ {10}$. We are looking for the number of shoppers on the first day, $ a$. Finding the first term $\begin{aligned} {1258}&={a} \cdot \dfrac{1-\left({1.05}\right)^{{10}}}{1-\left({1.05}\right)} \\\\ \dfrac{1-\left({1.05}\right)}{1-\left({1.05}\right)^{{10}}} \cdot {1258} &= {a} \\\\ 100.017 &\approx {a} \end{aligned}$ Answer There were $100$ shoppers at the mall on the first day.